LMFDB - Newform orbit 4225.2.a.ba

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4225,2,Mod(1,4225)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4225, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$4225 = 5^{2} \cdot 13^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	4225.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$33.7367948540$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{3} - x^{2} - 3x + 1$ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 65)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$ -expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $q$ -expansion are expressed in terms of a basis $1,\beta_1,\beta_2$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + ( - \beta_{2} - 1) q^{2} + (\beta_1 + 1) q^{3} + (\beta_{2} - \beta_1 + 2) q^{4} + ( - \beta_{2} - 2 \beta_1) q^{6} + (\beta_{2} + \beta_1 - 1) q^{7} + (3 \beta_1 - 4) q^{8} + (\beta_{2} + 3 \beta_1) q^{9}+ \cdots + (\beta_{2} + 8 \beta_1) q^{99}+O(q^{100})$ q + (-b2 - 1) * q^2 + (b1 + 1) * q^3 + (b2 - b1 + 2) * q^4 + (-b2 - 2b1) q^6 + (b2 + b1 - 1) * q^7 + (3b1 - 4) q^8 + (b2 + 3b1) q^9 + (b2 + 2) * q^11 + (b1 - 1) * q^12 + (b2 - b1 - 1) * q^14 + (2b2 - 4b1 + 3) * q^16 + (-2b2 + 2) q^17 - 5b1 q^18 + b2 * q^19 + (2b2 + 2b1) * q^21 + (-2b2 + b1 - 5) q^22 + (-b1 + 5) * q^23 + (3b2 + 2b1 + 2) * q^24 + (4b2 + 4b1 + 2) * q^27 + (-b2 + b1 - 1) * q^28 + (-3b2 - 3b1 + 3) * q^29 + (-b2 - 2b1 + 4) q^31 + (-3b2 + 4b1 - 5) * q^32 + (b2 + 3b1 + 1) q^33 + (-2b2 - 2b1 + 4) * q^34 + (-2b2 + 4b1 - 5) * q^36 + (b2 + 3b1 - 1) q^37 + (b1 - 3) * q^38 + (2b2 - 2b1 + 2) * q^41 + (-2b1 - 4) q^42 + (-2b2 + 3b1 + 1) * q^43 + (3b2 - 4b1 + 8) * q^44 + (-5b2 + 2b1 - 6) * q^46 + (-b2 - b1 - 3) * q^47 + (-2b2 - 3b1 - 7) * q^48 + (-2b2 - 3) q^49 + (-2b2 + 4) q^51 + (4b2 + 2b1 + 2) * q^53 + (-2b2 - 4b1 - 10) * q^54 + (-b2 - b1 + 7) * q^56 + (b2 + b1 - 1) * q^57 + (-3b2 + 3b1 + 3) * q^58 + (-3b2 + 2b1 + 2) * q^59 + (b2 + 3b1 + 1) q^61 + (-4b2 + 3b1 - 3) * q^62 + (b2 + 3b1 + 5) q^63 + (b2 - 3b1 + 12) q^64 + (-b2 - 5b1 - 1) q^66 + (-5b2 - b1 - 3) q^67 + (2b1 - 4) q^68 + (-b2 + 3b1 + 3) q^69 + (-b2 + 6b1 + 2) q^71 + (5b2 + 15) q^72 + (-b2 - 3b1 + 9) q^73 + (b2 - 5b1 + 1) q^74 + (b2 - 2b1 + 4) q^76 + 2b1 q^77 + (4b2 + 2b1 - 6) * q^79 + (5b2 + 5b1 + 6) * q^81 + (-2b2 + 6b1 - 10) * q^82 + (b2 - b1 - 7) * q^83 + 2 * q^84 + (-b2 - 8b1 + 8) q^86 + (-6b2 - 6b1) * q^87 + (-4b2 + 9b1 - 11) * q^88 + (-4b2 - 4b1 - 2) * q^89 + (6b2 - 7b1 + 13) * q^92 + (-3b2 - b1 + 1) q^93 + (3b2 + b1 + 5) q^94 + (b2 + 6) * q^96 + (-6b2 - 4b1 + 6) * q^97 + (3b2 - 2b1 + 9) * q^98 + (b2 + 8b1) q^99
$\operatorname{Tr}(f)(q)$	$=$	$3 q - 3 q^{2} + 4 q^{3} + 5 q^{4} - 2 q^{6} - 2 q^{7} - 9 q^{8} + 3 q^{9} + 6 q^{11} - 2 q^{12} - 4 q^{14} + 5 q^{16} + 6 q^{17} - 5 q^{18} + 2 q^{21} - 14 q^{22} + 14 q^{23} + 8 q^{24} + 10 q^{27} - 2 q^{28}+ \cdots + 8 q^{99}+O(q^{100})$ 3 * q - 3 * q^2 + 4 * q^3 + 5 * q^4 - 2 * q^6 - 2 * q^7 - 9 * q^8 + 3 * q^9 + 6 * q^11 - 2 * q^12 - 4 * q^14 + 5 * q^16 + 6 * q^17 - 5 * q^18 + 2 * q^21 - 14 * q^22 + 14 * q^23 + 8 * q^24 + 10 * q^27 - 2 * q^28 + 6 * q^29 + 10 * q^31 - 11 * q^32 + 6 * q^33 + 10 * q^34 - 11 * q^36 - 8 * q^38 + 4 * q^41 - 14 * q^42 + 6 * q^43 + 20 * q^44 - 16 * q^46 - 10 * q^47 - 24 * q^48 - 9 * q^49 + 12 * q^51 + 8 * q^53 - 34 * q^54 + 20 * q^56 - 2 * q^57 + 12 * q^58 + 8 * q^59 + 6 * q^61 - 6 * q^62 + 18 * q^63 + 33 * q^64 - 8 * q^66 - 10 * q^67 - 10 * q^68 + 12 * q^69 + 12 * q^71 + 45 * q^72 + 24 * q^73 - 2 * q^74 + 10 * q^76 + 2 * q^77 - 16 * q^79 + 23 * q^81 - 24 * q^82 - 22 * q^83 + 6 * q^84 + 16 * q^86 - 6 * q^87 - 24 * q^88 - 10 * q^89 + 32 * q^92 + 2 * q^93 + 16 * q^94 + 18 * q^96 + 14 * q^97 + 25 * q^98 + 8 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $x^{3} - x^{2} - 3x + 1$ x^3 - x^2 - 3*x + 1 :

$\beta_{1}$	$=$	$\nu$ v
$\beta_{2}$	$=$	$\nu^{2} - \nu - 2$ v^2 - v - 2

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$\beta_1$ b1
$\nu^{2}$	$=$	$\beta_{2} + \beta _1 + 2$ b2 + b1 + 2

Display $\beta_i$ in terms of $\nu^j$

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label

\iota_m(\nu)

a_{2}

a_{3}

a_{4}

a_{5}

a_{6}

a_{7}

a_{8}

a_{9}

a_{10}

1.1

−1.48119
2.17009
0.311108

−2.67513

−0.481194

5.15633

1.28726

−0.806063

−8.44358

−2.76845

1.2

−1.53919

3.17009

0.369102

−4.87936

1.70928

2.51026

7.04945

1.3

1.21432

1.31111

−0.525428

1.59210

−2.90321

−3.06668

−1.28100

Atkin-Lehner signs

$p$	Sign
$5$	$-1$
$13$	$+1$

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	4225.2.a.ba		3
5.b	even	2	1		4225.2.a.bh		3
5.c	odd	4	2		845.2.b.c		6
13.b	even	2	1		325.2.a.k		3
39.d	odd	2	1		2925.2.a.bf		3
52.b	odd	2	1		5200.2.a.cb		3
65.d	even	2	1		325.2.a.j		3
65.f	even	4	2		845.2.d.b		6
65.h	odd	4	2		65.2.b.a	✓	6
65.k	even	4	2		845.2.d.a		6
65.o	even	12	4		845.2.l.e		12
65.q	odd	12	4		845.2.n.g		12
65.r	odd	12	4		845.2.n.f		12
65.t	even	12	4		845.2.l.d		12
195.e	odd	2	1		2925.2.a.bj		3
195.s	even	4	2		585.2.c.b		6
260.g	odd	2	1		5200.2.a.cj		3
260.p	even	4	2		1040.2.d.c		6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
65.2.b.a	✓	6	65.h	odd	4	2
325.2.a.j		3	65.d	even	2	1
325.2.a.k		3	13.b	even	2	1
585.2.c.b		6	195.s	even	4	2
845.2.b.c		6	5.c	odd	4	2
845.2.d.a		6	65.k	even	4	2
845.2.d.b		6	65.f	even	4	2
845.2.l.d		12	65.t	even	12	4
845.2.l.e		12	65.o	even	12	4
845.2.n.f		12	65.r	odd	12	4
845.2.n.g		12	65.q	odd	12	4
1040.2.d.c		6	260.p	even	4	2
2925.2.a.bf		3	39.d	odd	2	1
2925.2.a.bj		3	195.e	odd	2	1
4225.2.a.ba		3	1.a	even	1	1	trivial
4225.2.a.bh		3	5.b	even	2	1
5200.2.a.cb		3	52.b	odd	2	1
5200.2.a.cj		3	260.g	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(4225))$ :

T_{2}^{3} + 3T_{2}^{2} - T_{2} - 5

T_{3}^{3} - 4T_{3}^{2} + 2T_{3} + 2

T_{7}^{3} + 2T_{7}^{2} - 4T_{7} - 4

T_{11}^{3} - 6T_{11}^{2} + 8T_{11} + 2

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{3} + 3T^{2} - T - 5$ T^3 + 3*T^2 - T - 5
$3$	$T^{3} - 4 T^{2} + \cdots + 2$ T^3 - 4T^2 + 2T + 2
$5$	$T^{3}$ T^3
$7$	$T^{3} + 2 T^{2} + \cdots - 4$ T^3 + 2T^2 - 4T - 4
$11$	$T^{3} - 6 T^{2} + \cdots + 2$ T^3 - 6T^2 + 8T + 2
$13$	$T^{3}$ T^3
$17$	$T^{3} - 6 T^{2} + \cdots + 8$ T^3 - 6T^2 - 4T + 8
$19$	$T^{3} - 4T + 2$ T^3 - 4*T + 2
$23$	$T^{3} - 14 T^{2} + \cdots - 86$ T^3 - 14T^2 + 62T - 86
$29$	$T^{3} - 6 T^{2} + \cdots + 108$ T^3 - 6T^2 - 36T + 108
$31$	$T^{3} - 10 T^{2} + \cdots + 26$ T^3 - 10T^2 + 20T + 26
$37$	$T^{3} - 28T - 52$ T^3 - 28*T - 52
$41$	$T^{3} - 4 T^{2} + \cdots - 32$ T^3 - 4T^2 - 32T - 32
$43$	$T^{3} - 6 T^{2} + \cdots + 278$ T^3 - 6T^2 - 46T + 278
$47$	$T^{3} + 10 T^{2} + \cdots + 20$ T^3 + 10T^2 + 28T + 20
$53$	$T^{3} - 8 T^{2} + \cdots + 304$ T^3 - 8T^2 - 40T + 304
$59$	$T^{3} - 8 T^{2} + \cdots + 262$ T^3 - 8T^2 - 40T + 262
$61$	$T^{3} - 6 T^{2} + \cdots - 4$ T^3 - 6T^2 - 16T - 4
$67$	$T^{3} + 10 T^{2} + \cdots - 604$ T^3 + 10T^2 - 60T - 604
$71$	$T^{3} - 12 T^{2} + \cdots + 754$ T^3 - 12T^2 - 88T + 754
$73$	$T^{3} - 24 T^{2} + \cdots - 236$ T^3 - 24T^2 + 164T - 236
$79$	$T^{3} + 16 T^{2} + \cdots - 16$ T^3 + 16T^2 + 24T - 16
$83$	$T^{3} + 22 T^{2} + \cdots + 316$ T^3 + 22T^2 + 152T + 316
$89$	$T^{3} + 10 T^{2} + \cdots - 200$ T^3 + 10T^2 - 52T - 200
$97$	$T^{3} - 14 T^{2} + \cdots + 200$ T^3 - 14T^2 - 84T + 200
show more
show less

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

$n$ :		e.g. 2-40 or 990-1000
Significant digits:
Format:

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more